---
title: "Fountain clocks: the density ratio is part of the correction"
description: "When can two atom-number settings identify the zero-density frequency, and when do they hide a bias?"
canonical_url: "https://sandboxphysics.com/stories/quantum-evidence/fountain-clock-collision-density-extrapolation/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/quantum-evidence/fountain-clock-collision-density-extrapolation/"
last_modified: "2026-10-18"
---

[Quantum physics: reading the evidence](https://sandboxphysics.com/stories/quantum-evidence/)

Nobel A · Evidence and control

# Fountain clocks: the density ratio is part of the correction

Sandbox Physics · 2026-10-18

When can two atom-number settings identify the zero-density frequency, and when do they hide a bias?

## More atoms change two things

More independent detected atoms can reduce population-counting uncertainty. In a cold cesium fountain, interactions between atoms can also shift the measured frequency. Improving counting statistics therefore creates an experimental tradeoff if density-dependent corrections are not known well enough.

Pereira Dos Santos and colleagues addressed this problem in 2002 by controlling population transfer before Ramsey interrogation. The key was to change the number of atoms while preserving the spatial and velocity distributions closely enough that the ratio of effective collision densities could be known.

Read the evidence: F. Pereira Dos Santos et al. (2002), Controlling the cold collision shift in high precision atomic interferometry — arXiv:physics/0209010v1 S. Bize et al. (2005), Cold atom Clocks and Applications — author manuscript, clock operation and collision control

## Write the extrapolation assumption explicitly

Assume the measured frequency depends linearly on effective density with one common coefficient. Label the low- and high-density frequencies accordingly, and let the ratio be the low effective density divided by the high one. Eliminating the common coefficient gives the zero-density intercept below.

The ratio lies strictly between zero and one. It is an effective collision-density ratio, not automatically the ratio of fluorescence counts. To use a detected-number ratio instead, preparation and detection must preserve the relevant relation between counted atoms and the density sampled during interrogation.

$$
\nu_0=\frac{\nu_{\mathrm L}-r\nu_{\mathrm H}}{1-r},\qquad r=\frac{n_{\mathrm L}}{n_{\mathrm H}}
$$

Read the evidence: F. Pereira Dos Santos et al. (2002), Controlling the cold collision shift in high precision atomic interferometry — arXiv:physics/0209010v1

## An incorrect ratio moves the intercept

Choose invented frequency offsets relative to an arbitrary reference: plus 0.4 millihertz at high density and plus 0.2 millihertz at half that density. The linear extrapolation returns zero offset. The cancellation can be checked directly from the two underlying density points.

Now keep those same measured offsets but mistakenly use a density ratio of 0.55. The inferred intercept becomes about minus 0.044 millihertz. Taking more data at the same two preparations can reduce their counting scatter while leaving this ratio error in the answer.

Read the evidence: F. Pereira Dos Santos et al. (2002), Controlling the cold collision shift in high precision atomic interferometry — arXiv:physics/0209010v1

## Half a population need not mean half the density

Changing loading time can alter cloud size and temperature as well as atom number. Changing an ordinary selection-pulse area can weight trajectories differently if the microwave field is nonuniform. Two samples with a simple detected-count ratio may then have different collision coefficients.

The published method used full and interrupted adiabatic passages, followed by removal of unwanted-state atoms, to make selection more uniform. Its measured number ratios were near 0.506, not an instruction to replace every measured ratio by exactly one half. The preparation comparison and its residual limitations belong in the correction.

Read the evidence: F. Pereira Dos Santos et al. (2002), Controlling the cold collision shift in high precision atomic interferometry — arXiv:physics/0209010v1

## Extrapolation also amplifies noise

At an exact one-half density ratio, the intercept is twice the low-density frequency minus the high-density frequency. If those two estimates have independent equal standard uncertainty, the intercept’s standard uncertainty is larger by the square root of five. Shared oscillator noise instead requires their covariance.

The denominator also makes ratios near one poorly conditioned: nearly equal densities supply little extrapolation leverage. More than two densities can test curvature and changes of preparation. Two points alone always define a line, so they cannot establish that a linear density law is adequate.

Read the evidence: F. Pereira Dos Santos et al. (2002), Controlling the cold collision shift in high precision atomic interferometry — arXiv:physics/0209010v1

## A correction is a measured procedure

The Ramsey Lab illustrates finite-count interrogation and the Rabi Lab illustrates population transfer. Neither models a colliding cesium fountain or performs the historical adiabatic-density correction. Their records should not be combined and labeled a new experimental uncertainty budget.

For the real inference, retain the frequency difference, number-ratio calibration, preparation settings and tests of the common-coefficient assumption. The result is a zero-density estimate under those conditions, while the atoms used to obtain it were measured at finite density. That distinction is what makes the correction reviewable.

Read the evidence: F. Pereira Dos Santos et al. (2002), Controlling the cold collision shift in high precision atomic interferometry — arXiv:physics/0209010v1 S. Bize et al. (2005), Cold atom Clocks and Applications — author manuscript, clock operation and collision control

## Continue with the models

These are existing teaching Labs. Their simulated records are separate from the historical observations discussed above.

- [Compare finite-count frequency estimates](https://sandboxphysics.com/labs/ramsey-frequency/)
- [Inspect controlled population transfer](https://sandboxphysics.com/labs/rabi-pulse-calibration/)

## Sources and reading notes

We read Pereira Dos Santos et al.’s four-page author manuscript, arXiv:physics/0209010v1, submitted in 2002, including its preparation comparison and Tables I–II. The PDF contains a later typesetting date; the submission record and linked PRL identify the 2002 work. All offsets and uncertainty examples below are invented, not a reanalysis of its fountain data.

1. [F. Pereira Dos Santos et al. (2002), Controlling the cold collision shift in high precision atomic interferometry — arXiv:physics/0209010v1](https://arxiv.org/pdf/physics/0209010)
2. [S. Bize et al. (2005), Cold atom Clocks and Applications — author manuscript, clock operation and collision control](https://arxiv.org/pdf/physics/0502117)

[Blackbody radiation: a whole curve, not one quantum](https://sandboxphysics.com/stories/quantum-evidence/blackbody-spectrum-planck-wien/) [Photoelectric evidence: the voltage is not yet the energy](https://sandboxphysics.com/stories/quantum-evidence/photoelectric-stopping-potential-contact/) [Oil drops: finding a charge unit in falling times](https://sandboxphysics.com/stories/quantum-evidence/millikan-oil-drop-charge-inference/) [Atomic spectra: evidence for states, not photographs of orbits](https://sandboxphysics.com/stories/quantum-evidence/atomic-spectra-stationary-states/) [Franck–Hertz: what a falling current can establish](https://sandboxphysics.com/stories/quantum-evidence/franck-hertz-discrete-excitation/) [Compton scattering: light must balance momentum too](https://sandboxphysics.com/stories/quantum-evidence/compton-scattering-energy-momentum/) [de Broglie: a wavelength for matter, a speed for the packet](https://sandboxphysics.com/stories/quantum-evidence/de-broglie-matter-waves-phase-packets/) [Matrices and waves: two routes to the same predictions](https://sandboxphysics.com/stories/quantum-evidence/matrix-wave-mechanics-observables/) [Born: how a scattering direction became a probability](https://sandboxphysics.com/stories/quantum-evidence/born-scattering-probability/) [Pauli: count the states before filling the shells](https://sandboxphysics.com/stories/quantum-evidence/pauli-spectra-exclusion-principle/) [Dirac’s electron: a successful equation with an unfinished story](https://sandboxphysics.com/stories/quantum-evidence/dirac-electron-spin-relativity/) [Zeeman lines: polarization belongs to the evidence](https://sandboxphysics.com/stories/quantum-evidence/zeeman-lines-polarization-lorentz/) [The Stark effect: an electric field changes the spectrum](https://sandboxphysics.com/stories/quantum-evidence/stark-effect-electric-field-spectra/) [Canal rays: moving light does not identify the emitter by itself](https://sandboxphysics.com/stories/quantum-evidence/stark-canal-rays-doppler-emitters/) [Stern–Gerlach: two traces, then a new interpretation](https://sandboxphysics.com/stories/quantum-evidence/stern-gerlach-silver-beam-spin/) [Stern’s proton moment: a small deflection with a large consequence](https://sandboxphysics.com/stories/quantum-evidence/stern-proton-moment-molecular-hydrogen/) [Magnetic resonance: three ways to notice a transition](https://sandboxphysics.com/stories/quantum-evidence/magnetic-resonance-beams-absorption-induction/) [Optical pumping: make a population difference visible](https://sandboxphysics.com/stories/quantum-evidence/optical-pumping-populations-double-resonance/) [The Lamb shift: a missing signal reveals a missing correction](https://sandboxphysics.com/stories/quantum-evidence/lamb-shift-microwave-evidence/) [Kusch and Foley: a small mismatch between magnetic factors](https://sandboxphysics.com/stories/quantum-evidence/kusch-foley-electron-magnetic-moment/) [Mössbauer resonance: recoil has not disappeared](https://sandboxphysics.com/stories/quantum-evidence/mossbauer-recoil-free-resonance/) [Ramsey and ion traps: make time to measure](https://sandboxphysics.com/stories/quantum-evidence/ramsey-ion-traps-measurement/) [Ramsey clocks: a stable fringe can carry a phase bias](https://sandboxphysics.com/stories/quantum-evidence/ramsey-cavity-phase-frequency-bias/) [Quantum projection noise: better detection still leaves a random count](https://sandboxphysics.com/stories/quantum-evidence/quantum-projection-noise-population-counts/) [Paul’s trap: stability comes from the changing field](https://sandboxphysics.com/stories/quantum-evidence/paul-trap-dynamic-stability-micromotion/) [Dehmelt’s single electron: spin appears as an axial frequency shift](https://sandboxphysics.com/stories/quantum-evidence/dehmelt-single-electron-penning-spin-readout/) [Laser cooling: when the temperature contradicted the model](https://sandboxphysics.com/stories/quantum-evidence/laser-cooling-below-doppler-limit/) [Zeeman slowing: keep a moving atom within reach of the light](https://sandboxphysics.com/stories/quantum-evidence/zeeman-slowing-neutral-atom-beam/) [Chu’s optical molasses: slowing motion while atoms still escape](https://sandboxphysics.com/stories/quantum-evidence/chu-optical-molasses-diffusion-escape/) [Cold-atom thermometry: read the flight, then infer the temperature](https://sandboxphysics.com/stories/quantum-evidence/cold-atom-ballistic-thermometry/) [Sisyphus cooling: where the energy leaves the hill](https://sandboxphysics.com/stories/quantum-evidence/sisyphus-cooling-polarization-energy/) [Subrecoil cooling: two narrow peaks are not zero motion](https://sandboxphysics.com/stories/quantum-evidence/subrecoil-cooling-dark-state-momentum/) [A dark interval from one ion: reading electron shelving](https://sandboxphysics.com/stories/quantum-evidence/electron-shelving-single-ion-jumps/) [Unequal sidebands: how cold is a trapped ion?](https://sandboxphysics.com/stories/quantum-evidence/sideband-asymmetry-ground-state-cooling/) [Quantum logic spectroscopy: let another ion report the answer](https://sandboxphysics.com/stories/quantum-evidence/quantum-logic-spectroscopy-shared-motion/) [Single quantum systems: what survives a measurement?](https://sandboxphysics.com/stories/quantum-evidence/single-quantum-measurement-backaction/) [A cavity cat loses coherence before it loses its energy](https://sandboxphysics.com/stories/quantum-evidence/cavity-cat-coherence-energy-decay/) [Counting stored light: one atom gives a clue](https://sandboxphysics.com/stories/quantum-evidence/qnd-photon-counting-atomic-records/) [Quantum feedback: a target needs more than an estimate](https://sandboxphysics.com/stories/quantum-evidence/quantum-feedback-photon-number-verification/) [Bell tests: correlations need an experimental contract](https://sandboxphysics.com/stories/quantum-evidence/bell-tests-correlations-loopholes/) [Freedman–Clauser: the denominator belongs to the experiment](https://sandboxphysics.com/stories/quantum-evidence/freedman-clauser-coincidence-normalization/) [Aspect’s switches: read the spacetime before the slogan](https://sandboxphysics.com/stories/quantum-evidence/aspect-switching-spacetime-records/) [An event-ready Bell test: count the trials before the wins](https://sandboxphysics.com/stories/quantum-evidence/event-ready-bell-trials-finite-statistics/) [Photon Bell tests: the trials with no clicks still count](https://sandboxphysics.com/stories/quantum-evidence/photon-bell-losses-optical-trials/) [GHZ correlations: four settings, no consistent instruction table](https://sandboxphysics.com/stories/quantum-evidence/ghz-parity-three-photon-records/) [Teleportation: read the conditions before the coincidence](https://sandboxphysics.com/stories/quantum-evidence/quantum-teleportation-conditional-records/) [Heralded photons: the missing coincidences need a denominator](https://sandboxphysics.com/stories/quantum-evidence/heralded-photon-anticorrelation-gates/) [The Hong–Ou–Mandel dip: which alternatives became indistinguishable?](https://sandboxphysics.com/stories/quantum-evidence/hong-ou-mandel-dip-mode-overlap/) [Entanglement swapping: four clicks define the selected pair](https://sandboxphysics.com/stories/quantum-evidence/entanglement-swapping-fourfold-records/) [Delayed-choice swapping: later labels, earlier records](https://sandboxphysics.com/stories/quantum-evidence/delayed-choice-swapping-sorted-records/) [Dense coding: three messages and a resource ledger](https://sandboxphysics.com/stories/quantum-evidence/dense-coding-three-messages-resources/) [Photon tomography: a physical matrix still needs calibrated counts](https://sandboxphysics.com/stories/quantum-evidence/photon-tomography-counts-physical-states/)